On the minimum distance of ternary cyclic codes

There are many ways to find lower bounds for the minimum distance of a cyclic code, based on investigation of the defining set. Some new theorems are derived. These and earlier techniques are applied to find lower bounds for the minimum distance of ternary cyclic codes. Furthermore, the exact minimum distance of ternary cyclic codes of length less than 40 is computed numerically. A table is given containing all ternary cyclic codes of length less than 40 and having a minimum distance exceeding the BCH bound. It seems that almost all lower bounds are equal to the minimum distance. Especially shifting, which is also done by computer, seems to be very powerful. For length 40⩽n⩽50, only lower bounds are computed. In many cases (derived theoretically), however, these lower bounds are equal to the minimum distance     

A new lower bound for the minimum distance of a cyclic code

A new lower bound for the minimum distance of a linear code is derived. When applied to cyclic codes both the Bose-Chaudhuri-Hocquenghem (BCH) bound and the Hartmann-Tzeng (HT) bound are obtained as corollaries. Examples for which the new bound is superior to these two bounds, as well as to the Carlitz-Uchiyama bound, are given.     

On Randomized Linear Network Codes and Their Error Correction Capabilities

Randomized linear network code for single source multicast was introduced and analyzed in Ho et al. (IEEE Transactions on Information Theory, October 2006) where the main results are upper bounds for the failure probability of the code. In this paper, these bounds are improved and tightness of the new bounds is studied by analyzing the limiting behavior of the failure probability as the field size goes to infinity. In the linear random coding setting for single source multicast, the minimum distance of the code defined in Zhang, (IEEE Transactions on Information Theory, January 2008) is a random variable taking nonnegative integer values that satisfy the inequality in the Singleton bound recently established in Yeung and Cai (Communications in Information and Systems, 2006) for network error correction codes. We derive a bound on the probability mass function of the minimum distance of the random linear network code based on our improved upper bounds for the failure probability. Codes having the highest possible minimum distance in the Singleton bound are called maximum distance separable (MDS). The bound on the field size required for the existence of MDS codes reported in Zhang, (IEEE Transactions on Information Theory, January 2008) and Matsumoto (arXiv:cs.IT/0610121, Oct. 2006) suggests that such codes exist only when field size is large. Define the degradation of a code as the difference between the highest possible minimum distance in the Singleton bound and the actual minimum distance of the code. The bound for the probability mass function of the minimum distance leads to a bound on the field size required for the existence of network error correction codes with a given maximum degradation. The result shows that allowing minor degradation reduces the field size required dramatically.     

Minimum distance bounds for cyclic codes and Deligne's theorem

At the present time, there are very good methods to obtain bounds for the minimum distance of BCH codes and their duals. On the other hand, there are few other bounds suitable for general cyclic codes. Therefore, research Problem 9.9 of MacWilliams and Sloane (1977), The Theory of Error-Correcting Codes, asks if the bound of Deligne (1974) for exponential sums in several variables or the bound of Lang and Weil (1954), can be used to obtain bounds on the minimum distance of codes. This question is answered in the affirmative by showing how Deligne's theorem can be made to yield a lower bound on the minimum distance of certain classes of cyclic codes. In the process, an infinite family of binary cyclic codes is presented for which the bound on minimum distance so derived is as tight as possible. In addition, an infinite family of polynomials of degree 3 in 2 variables over a field of characteristic 2, for which Deligne's bound is tight, is exhibited. Finally, a bound is presented for the minimum distance of the duals of the binary subfield subcodes of generalized Reed-Muller codes as well as for the corresponding cyclic codes. It is noted that these codes contain examples of the best binary cyclic codes     

On the minimum distance of cyclic codes

The main result is a new lower bound for the minimum distance of cyclic codes that includes earlier bounds (i.e., BCH bound, HT bound, Roos bound). This bound is related to a second method for bounding the minimum distance of a cyclic code, which we call shifting. This method can be even stronger than the first one. For all binary cyclic codes of length< 63(with two exceptions), we show that our methods yield the true minimum distance. The two exceptions at the end of our list are a code and its even-weight subcode. We treat several examples of cyclic codes of lengthgeq 63.     

Tailbiting codes obtained via convolutional codes with large active distance-slopes

The slope of the active distances is an important parameter when investigating the error-correcting capability of convolutional codes and the distance behavior of concatenated convolutional codes. The slope of the active distances is equal to the minimum average weight cycle in the state-transition diagram of the encoder. A general upper bound on the slope depending on the free distance of the convolutional code and new upper bounds on the slope of special classes of binary convolutional codes are derived. Moreover, a search technique, resulting in new tables of rate R=1/2 and rate R=1/3 convolutional encoders with high memories and large active distance-slopes is presented. Furthermore, we show that convolutional codes with large slopes can be used to obtain new tailbiting block codes with large minimum distances. Tables of rate R=1/2 and rate R=1/3 tailbiting codes with larger minimum distances than the best previously known quasi-cyclic codes are given. Two new tailbiting codes also have larger minimum distances than the best previously known binary linear block codes with same size and length. One of them is also superior in terms of minimum distance to any previously known binary nonlinear block code with the same set of parameters.     

Bounds on distance distributions in codes of known size

We treat the problem of bounding components of the possible distance distributions of codes given the knowledge of their size and possibly minimum distance. Using the Beckner inequality from harmonic analysis, we derive upper bounds on distance distribution components which are sometimes better than earlier ones due to Ashikhmin, Barg, and Litsyn. We use an alternative approach to derive upper bounds on distance distributions in linear codes. As an application of the suggested estimates we get an upper bound on the undetected error probability for an arbitrary code of given size. We also use the new bounds to derive better upper estimates on the covering radius, as well as a lower bound on the error-probability threshold, as a function of the code's size and minimum distance.     

Bounds for Codes in Products of Spaces, Grassmann, and Stiefel Manifolds

Upper bounds are derived for codes in Stiefel and Grassmann manifolds with given minimum chordal distance. They stem from upper bounds for codes in the product of unit spheres and projective spaces. The new bounds are asymptotically better than the previously known ones.      

A strengthened asymptotic Gilbert bound for convolutional codes (Corresp.)

The usual statement of the Gilbert bound is an assertion that at least one member of a given ensemble of codes satisfies a minimum distance criterion. This result is strengthened by showing that for sufficiently large constraint lengths, an arbitrarily large fraction of the ensemble of codes have minimum distance exceeding the usual asymptotic Gilbert bound. New asymptotic bounds are derived for the nonbinary case.     

Upper bounds on the rate of LDPC codes as a function of minimum distance

New upper bounds on the rate of low-density parity-check (LDPC) codes as a function of the minimum distance of the code are derived. The bounds apply to regular LDPC codes, and sometimes also to right-regular LDPC codes. Their derivation is based on combinatorial arguments and linear programming. The new bounds improve upon the previous bounds due to Burshtein et al. It is proved that at least for high rates, regular LDPC codes with full-rank parity-check matrices have worse relative minimum distance than the one guaranteed by the Gilbert-Varshamov bound.     

Combinatorial analysis of the minimum distance of turbo codes

In this paper, new upper bounds on the maximum attainable minimum Hamming distance of turbo codes with arbitrary-including the best-interleavers are established using a combinatorial approach. These upper bounds depend on the interleaver length, the code rate, and the scramblers employed in the encoder. Examples of the new bounds for particular turbo codes are given and discussed. The new bounds are tighter than all existing ones and prove that the minimum Hamming distance of turbo codes cannot asymptotically grow at a rate more than the third root of the codeword length     

Free distance bounds for convolutional codes

The best asymptotic bounds presently known on free distance for convolutional codes are presented from a unified point of view. Upper and lower bounds for both time-varying and fixed codes are obtained. A comparison is made between bounds for nonsystematic and systematic codes which shows that more free distance is available with nonsystematic codes. This result is important when selecting codes for use with sequential or maximum-likelihood (Viterbi) decoding since the probability of decoding error is closely related to the free distance of the code. An ancillary result, used in proving the lower bound on free distance for time-varying nonsystematic codes, furnishes a generalization of two earlier bounds on the definite decoding minimum distance of convolutional codes.     

Source and channel rate allocation for channel codes satisfying theGilbert-Varshamov or Tsfasman-Vladut-Zink bounds

We derive bounds for optimal rate allocation between source and channel coding for linear channel codes that meet the Gilbert-Varshamov or Tsfasman-Vladut-Zink (1984) bounds. Formulas giving the high resolution vector quantizer distortion of these systems are also derived. In addition, we give bounds on how far below the channel capacity the transmission rate should be for a given delay constraint. The bounds obtained depend on the relationship between channel code rate and relative minimum distance guaranteed by the Gilbert-Varshamov bound, and do not require sophisticated decoding beyond the error correction limit. We demonstrate that the end-to-end mean-squared error decays exponentially fast as a function of the overall transmission rate, which need not be the case for certain well-known structured codes such as Hamming codes     

Bounds for Binary Codes With Narrow Distance Distributions

New lower bounds are presented on the second moment of the distance distribution of binary codes, in terms of the first moment of the distribution. These bounds are used to obtain upper bounds on the size of codes whose maximum distance is close to their minimum distance. It is then demonstrated how such bounds can be applied to bound from below the smallest attainable ratio between the maximum distance and the minimum distance of codes. Finally, counterparts of the bounds are derived for the special case of constant-weight codes.     

Upper bounds on codes correcting asymmetric errors (Corresp.)

A Survey is given of known upper bounds on codes correcting asymmetric errors. The bounds are improved by introducing new Ideas. By solving a linear programming problem an upper bound is given that is easy to compute for all codelengths and all minimum asymmetric distances.     

A note on nonlinear Xing codes

Nonlinear Xing codes are considered. It is shown that Xing codes of length p-1 (where p is a prime) are subcodes of cosets of Reed-Solomon codes whose minimum distance equals Xing's lower bound on the minimum distance. This provides a straightforward proof for the lower bound on the minimum distance of the codes. The alphabet size of Xing codes is restricted not to be larger than the characteristic of the relevant finite field F/sub r/. It is shown that codes with the same length and the same lower bounds on the size and minimum distance as Xing codes exist for any alphabet size not exceeding the size r of the relevant finite field, thus extending Xing's results.     

Six new binary quasi-cyclic codes

Six new quasi-cyclic codes are presented, which improve the lower bounds on the minimum distance for a binary code. A local exhaustive search is used to find these codes and many other quasi-cyclic codes which attain the lower bounds.<>     

Performance of low-density parity-check codes with linear minimum distance

This correspondence studies the performance of the iterative decoding of low-density parity-check (LDPC) code ensembles that have linear typical minimum distance and stopping set size. We first obtain a lower bound on the achievable rates of these ensembles over memoryless binary-input output-symmetric channels. We improve this bound for the binary erasure channel. We also introduce a method to construct the codes meeting the lower bound for the binary erasure channel. Then, we give upper bounds on the rate of LDPC codes with linear minimum distance when their right degree distribution is fixed. We compare these bounds to the previously derived upper bounds on the rate when there is no restriction on the code ensemble.     

Distance-spectrum formulas on the largest minimum distance of blockcodes

A general formula for the asymptotic largest minimum distance (in block length) of deterministic block codes under generalized distance functions (not necessarily additive, symmetric, and bounded) is presented. As revealed in the formula, the largest minimum distance can be fully determined by the ultimate statistical characteristics of the normalized distance function evaluated under a properly chosen random-code generating distribution. Interestingly, the new formula has an analogous form to the general information-spectrum expressions of the channel capacity and the optimistic channel capacity, respectively derived by Verdu and Han (1994) and Chen and Alajaji (1998, 1999). As a result, a minor class of distance functions for which the largest minimum distance can be derived is characterized. A general Varshamov-Gilbert lower bound is next addressed. Some discussions on the tightness of the general Varshamov-Gilbert bound are also provided. Finally, lower bounds on the largest minimum distances for several specific block coding schemes are rederived in terms of the new formulas, followed by comparisons with the known results devoted to the same codes     

An upper bound on the minimum distance of a convolutional code

An upper bound on the minimum distance of a linear convolutional code is given which reduces to the Plotkin bound for the block code case. It is shown that most linear convolutional codes have a minimum distance strictly less than their average distance. A table of the bound for several rates is given for binary codes as well as a comparison with the known optimum values for codes of block length2.